An evil troll once captured a bunch of gnomes and told them, "Tomorrow, I will make you stand in a file, ordered by height such that a gnome can see exactly those gnomes that are shorter than him. I will place either a white cap or a black cap on each head. Then, starting from the tallest, each gnome has to declare aloud what he thinks the color of his own cap is. In the end, those who were correct will be spared; others will be eaten, silently." The gnomes set thinking and came up with a strategy. How many of them survived?